# The Enigma Machine and How It Worked

These substitutions occur because of the positions of the rotors on the inside of the machine. The rotors are first situated at any random letter in the alphabet and when the button is pressed, the letter that is at the beginning of the rotor is put onto the paper. Then if the same letter were to be pressed again, a different letter would appear because the rotor will have changed position. The fully typed and encoded message would be printed out and mailed to the person it was designated to and then that person would put in that code into their enigma machine and that machine would decipher the encoded message.

Why did the Enigma machine work? : The enigma machine worked so well because there were so many different combinations of rotors that could have been used because each rotor had a different arbitrary sorting of letters. There were so many combinations for the enigma machine and its rotors.

Things became much more complicated when German scientists began to use 5 different rotors which could fit into only 3 slots, which increased the amount of combinations that could be used in the enigma machine.

With all these rotors and so few slots that were used in the enigma machine, there were countless possibilities of combinations to choose from. When people were decrypting the codes they were sent, they had to make sure they put the rotors in the exact same position as the person who sent it. This would increase the amount of possibilities that the enigma machine possessed which made it a lot harder to crack because of all the possible combinations. My Goal: My goal in this paper is to answer several questions and provide my roof for these questions. I will be asking: 1. How many different ways can we fit 3 different rotors into 3 different slots? 2. How many combinations can we find if we have 4 different rotors that will fit into four slots? 3. How many combinations can I find if I have 5 rotors that can go into five slots? 4. How many combinations are there if I have 5 rotors but only three slots? 5. And finally, if I have 26 different starting places for each dial, how many starting places would I have in total?

Note: I will also be attempting to show many patterns that arise while searching for the answers to these questions and I will try to explain these patterns. List of Symbols: Before I begin, I feel it is important to list all the symbols that I will be using. I will be using the numbers: 1, 2, 3, 4, and 5 to represent a rotor, with 1 representing the first rotor, 2 representing the second rotor, and so on. Rotors that I will be using Rotors that I will be using Amount of Rotors, slots, and combos Amount of Rotors, slots, and combos

How will I proceed with this: I will first write out the amount of rotors, slots and combinations in a format like this: โX rotors: Y Slots: Z combinationsโ. Then underneath this general information I will write which rotors I will be using i. e. Rotors: 1/2/3. Then, I will show how I came up with the number of combinations. By looking at the amount of rotors and by representing each rotor with a number, i. e. 1, 2, 3, 4, or 5, I will be able to display all the possible combinations of the rotors. And then, underneath my combinations, I will write my explanation and, if there is one, the equation.

A way we can visualize why this works is that if we were to take a rock that had 2 scratches on it, then take another scratch that had 2 scratches on it, and placed them next to each other, then in total there would be 4 scratches when the rocks were in that certain position. But if we were to arrange the rocks so that the rock that was in front of the other rock was now in the back, and the rock that was in the back was now in the front, we have a whole new position for the rocks that still have 4 scratches on them. They are the same scratches. Changing the position of the rock is like putting each rotor into a different slot every time, the rotors will not always be in the same slot every single time, they are in different positions every time, by taking into account the amount of slots, one can calculate how many positions several rotors can take.

The idea of putting the rotors into different positions is the same idea of changing the position of the rock. By looking at how many rocks we have, we can figure out how many positions we can put them in. this is combinations. If we have 3 rocks, we can arrange them into 6 different positions. If we have 4 rocks, we can arrange each of them into 24 positions. This is why we use the amount of combinations of the rotors to figure out how many combinations are there for the dials in total. If we can multiply the amount of combinations by the amount of dials, we will figure out all the combinations for the dials. Equation: (Amount of combinations for rotors)(amount of dials on rotor)=Amount of total amount of combinations for dials.